A note on the Venice lemma in differential \(K\)-theory
Among the models of differential \(K\)-theory are a model defined by triples in which a differential form is part of the cocycle data and a model that does not rely on that differential form. The former model is due to [\textit{D. S. Freed} and \textit{J. Lott}, Geom. Topol. 14, No. 2, 903--966 (2010; Zbl 1197.58007)] and [\textit{K. R. Klonoff}, An index theorem in differential \(K\)-theory, Ph.D. thesis, University of Texas (2008)]. The latter model is due to [\textit{J. Simons} and \textit{D. Sullivan}, Clay Math. Proc. 11, 579--599 (2010; Zbl 1216.19009)], where the ``Venice lemma'' showed why dependence on the differential form was unnecessary. The theories defined by the two models are abstractly isomorphic. [\textit{M.-H. Ho}, Ann. Global Anal. Geom. 42, No. 4, 523--535 (2012; Zbl 1257.19005)] defined an explicit isomorphism. The paper under review gives a new proof of Ho's result by using the Venice lemma and a result of [\textit{T. Tradler} et al., J. \(K\)-Theory 12, No. 2, 331--361 (2013; Zbl 1327.19014)] to prove a lemma that states in simple terms that a class defined by a triple consisting of a product bundle, connection, and differential form is equivalent to the difference of classes defined by two such triples in which the differential forms are zero. The author's interest in twisted (and other) versions of differential \(K\)-theory motivated his search for a simpler proof of Ho's result.
scientific article; zbMATH DE number 7489064
scientific article